Let $S$ and $T$ be sets of propositional formulae.

(a) What does it mean to say that $S$ is deductively closed? What does it mean to say that $S$ is consistent? Explain briefly why if $S$ is inconsistent then some finite subset of $S$ is inconsistent.

(b) We write $S \vdash T$ to mean $S \vdash t$ for all $t \in T$. If $S \vdash T$ and $T \vdash S$ we say $S$ and $T$ are equivalent. If $S$ is equivalent to a finite set $F$ of formulae we say that $S$ is finitary. Show that if $S$ is finitary then there is a finite set $R \subset S$ with $R \vdash S$.

(c) Now let $T_{0}, T_{1}, T_{2}, \ldots$ be deductively closed sets of formulae with

$$T_{0} \subset T_{1} \subset T_{2} \subset \cdots$$

Show that each $T_{i}$ is consistent.

Let $T=\bigcup_{i=0}^{\infty} T_{i}$. Show that $T$ is consistent and deductively closed, but that it is not finitary.